How Do You Calculate the Equation of an Hourglass Curve?

[ssj5harsh]

I need to find the equation of the curve of an hourglass. Known: diameter, height. Time taken for the sand to be completely emptied is 1 minute. I would like it if someone would tell me where or how to start. I can't think of anything.

Thanks in advance.

 

[HallsofIvy]

The curve of an hourglass? An hourglass can be many different shapes, starting with a cone. The information you give as "known" has nothing to do with the shape.

 

[SSj5NOTHARSH]

i was looking for the same thing today and came across this question

what i came up with was a combination of 8 functions for a 3d representation, although i am not certain if they are correct. also of note is that this is for it laying on its side. anyway, here goes:

x = y³ + z³ + 1, For x, 0 to h
-x = y³ + z³ + 1, For x, -h to 0
-x = -y³ + z³ + 1, For x, -h to 0 (?)
x = -y³ + z³ + 1, For x, 0 to h (?)
x = y³ + z³ - 1, For x, -h to 0
-x = y³ + z³ - 1, For x, 0 to h
-x = -y³ + z³ - 1, For x, 0 to h (?)
x = -y³ + z³ - 1, For x, -h to 0 (?)

where 2h = height of hourglass

from here it (should) should just be a matter of scaling, choosing of sand, and experimental trial and error to come up with the correct amount of the particularly chosen sand.

again, am not sure if this is correct. . . .

Best Regards,

 

[SSj5NOTHARSH]

whoops, let's try this again. . . .

f(y,z) = n(y³ + z³) + a; For y = 0, and positive y; For x, 0 to h
f(y,z) = n(-y³ - z³) - a; For negative y; For x, -h to 0
f(y,z) = n(y³ - z³) - a; For negative y; For x, -h to 0
f(y,z) = n(-y³ + z³) + a; For y = 0, and positive y; For x, 0 to h
f(y,z) = n(y³ + z³) - a; For negative y; For x, 0 to h
f(y,z) = n(-y³ - z³) + a; For y = 0, and positive y; For x, -h to 0
f(y,z) = n(y³ - z³) + a; For y = 0, and positive y; For x, -h to 0
f(y,z) = n(-y³ + z³) - a; For negative y; For x, 0 to h

where h = 1/2 hourglass height,
a = the cubed root of the hourglass annulous radius,
and
n = a height to width coefficient